# Questions tagged [lie-superalgebras]

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### Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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### Invariants of Lie superalgebras

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
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### Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
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### What does “$d$-generated algebra” mean? [closed]

I would like to know what does ”$d$-generated algebra” mean? In particular “two-generated algebra”? @KonstantinosKanakoglou @LSpice Here is an example of a paper concerning “one-generated algebras”: ...
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### Supercommutator and covariant derivation

Let $(E,\nabla)\to X$ be a vector bundle endowed with a connection and $f\in End(E)$ a bundle endomorphism. We can express the covariant derivative of $f$ using a commutator $$\nabla f=[\nabla,f].$$ ...
211 views

### Semisimple super Lie algebras

There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones. Question: How about ...
279 views

### Lie powers of a graded vector space and Klyachko's theorem

Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
84 views

### Reference request: superconformal algebras and representations

I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
173 views

### Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$: $$(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0$$ which defines the anti-...
124 views

### Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...